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Can someone explain to me how there can be different kinds of infinities? I was reading "the man who loved only numbers" and came across the concept of countable and uncountable infinities, but they're only words to me. Any help would be appreciated.
Different kinds of infinities?
[mathfactor][1] is one I listen to. Does anyone else have a recommendation? [1]: http://mathfactor.uark.edu/
List of interesting math podcasts?
I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grash what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
How can you prove that the square root of two is irrational?
What is your favorite online graphing tool?
I was reading up on the Fibonacci Sequence when I've noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly simple, but I reckon I can't find any formula for calculating a Fibonacci number based on it's position. Is there a ...
How are we able to calculate specific numbers in the Fibonacci Sequence?
Suppose no one every taught you the names for ordinary numbers. Then suppose that you and I agreed that we would trade one bushel of corn for each of my sheep. But there's a problem, we don't know how to count the bushels or the sheep! So what do we do? We form a "bijection" between the two sets. That's just fan...
I'm told by smart people that `0.999... = 1` and I believe them but is there a proof that explains why?
Does .99999... = 1?
How many Natural (Integers) Numbers could you count? There are infinitely many, yet you can count them. It's called [Countable Set][1]. How many Real Numbers are there? Infinitely as well (Since at least every Natural Number is Areal Number). Yet you won't be able to count them (Intuitively, Let's say you name a...
Given a the semi-major axis radius and a flattening factor, is it possible to calculate the semi-minor radius?
How do you calculate the semi-minor axis for an elipsoid?
By matrix-defined, I mean <pre> < a b c > x < d e f > = | i j k | det(| a b c |) | d e f | </pre> (instead of the definition of the product of the magnitudes multiplied by the sign of their angle, in the direction orthogonal) If I try cross producting two vectors with no `k` component, I get one...
Why is the matrix-defined Cross Product of two 3D vectors always orthogonal?
Well, I am not sure where you want to embed the graphs, but [Wolfram Alpha][1] is pretty handy for graphing. It has most of the features of Mathematica, can handle 3D functions, and fancy scaling and such. I highly recommend it. [1]: http://www.wolframalpha.com/input/?i=sin(x)
I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator. Can you recommend any?
Can you recommend a decent online or software calculator?
What I really don't like about all the above answers, is the underlying assumption that `1/3=0.3333...`, How do you know that?. It seems to me like assuming the something which is already known. A proof I really like is: 0.9999... x 10 = 9.999999... 0.9999... x 9 + 0.99999.... = 9.99999.... ...
What does it mean when you refer to .99999...? Symbols don't mean anything in particular until you've *defined what you mean by them*. In this case the definition is that you're taking the limit of .9, .99, .999, .9999, etc. What does it mean to say that limit is 1? Well, it means that no matter how small a numbe...
What length of rope should be used to tie a cow to a fence post of a *circular* field so that the cow can only graze half of the grass within that field?
> A *countably infinite* set is a set for which you can list the elements a<sub>1</sub>,a<sub>2</sub>,a<sub>3</sub>,... For example, the set of all integers is countably infinite since I can list its elements as follows 0,1,-1,2,-2,3,-3,... So is the set of rational numbers, but this is more difficult to see. Let's ...
As I've understood it, what I've learned as the dot product is just one of many possible "inner product spaces". Can someone explain this concept? When is it useful to define it as something other than the dot product?
What is an inner product space?
This is simply not clicking for me. I'm currently learning math during the summer vacation and I'm on the chapter for relations and functions. There are five properties for a relation: Reflexive - R -> R Symmetrical - R -> S ; S -> R Antisymmetrical - R -> S && (R -> R || S -> S) Asymmetrical - R -> S && !(R -...
How do the Properties of Relations work?
Given two points in 2D space how would you calculate an angle from p1 to p2? How would this change in 3D space?
What is an elliptic curve, and how are they used in cryptography?
You can see that there are infinitely many natural numbers 1, 2, 3, ..., and infinitely many real numbers, such as 0, 1, pi, etc. But are these two infinities the same? Well, suppose you have two sets of objects, e.g. people and horses, and you want to know if the number of objects in one set is the same as in the o...
What length of rope should be used to tie a cow to an **exterior fence post** of a *circular* field so that the cow can only graze half of the grass within that field? ***updated:*** To be clear: the cow should be tied to a post on the exterior of the field, not a post at the center of the field.
I've recently started reading about Quaternions, and I keep reading that for example they're used in computer graphics and mechanics calculations to calculate movement and rotation, but without real explanations of the benefits of using them. I'm wondering what exactly can be done with Quaternions that can't be done...
Real world uses of Quaternions?
This is simply not clicking for me. I'm currently learning math during the summer vacation and I'm on the chapter for relations and functions. There are five properties for a relation: Reflexive - R -> R Symmetrical - R -> S ; S -> R Antisymmetrical - R -> S && (R -> R || S -> S) Asymmetrical - R -> S &&...
Let's say I know 'X' is a Gaussian Variable. Moreover, I know 'Y' is a Gaussian Variable and Y=X+Z. Let's say all of the are Independent. How can I prove Z is a Gaussian Variable? It's easy to show the other way around (X, Z Orthogonal and Normal hence create a Gaussian Vector hence any Linear Combination of ...
Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
What purposes do the Dot and Cross products serve? Do you have any clear examples of when you would use them?
I'm talking in the range of 10-12 years old, but this question isn't limited to only that range. Do you have any advice on cool things to show kids that might spark their interest in spending more time with math? The difficulty for some to learn math can be pretty overwhelming. Do you have any teaching techniques th...
What are some good ways to get children excited about math?
Given two points, around an origin (0,0), in 2D space how would you calculate an angle from p1 to p2. How would this change in 3D space?
How to accurately calculate erf(x) with a computer?
I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
This is in relation to the Euler Problem 13 from www.ProjectEuler.net. Work out the first ten digits of the sum of the following one-hundred 50-digit numbers. 37107287533902102798797998220837590246510135740250 Now, this was my thinking: I can freely discard the last fourty digits and leave the last ten. ...
Faulty logic when summing large integers?
Since you're asking for a reference, perhaps this will do? Wolfram Mathworld says the problem was listed by Rado in 1925. The reference is on the problem description page, [here][1]. [1]: http://mathworld.wolfram.com/LionandManProblem.html
It is 10 o'clock, and I have a box. Inside the box is a ball marked 1. At 10:30, I will remove the ball marked 1, and add two balls, labeled 2 and 3. At 10:45, I will remove the balls labeled 2 and 3, and add 4 balls, marked 4, 5, 6, and 7. 7.5 minutes before 11, I will remove the balls labeled 4, 5, and 6, and add...
paradox: increasing sequence that goes to 0?
The argument against this is that 0.99999999... is "somewhat" less than 1. How much exactly? 1 - 0.999999... = Ξ΅ If the above is true, the following also must be true: 9 Γ— (1 - 0.999999...) = Ξ΅ Γ— 9 Let's calculate: 0.999... Γ— 9 = ----------- 8.1 81 8...
I keep seeing this symbol βˆ‡ around and I know enough to understand that it represents the term "gradient." But what is a gradient? When would I want to use one mathematically?
What are gradients and how would I use them?
The argument against this is that 0.99999999... is "somewhat" less than 1. How much exactly? 1 - 0.999999... = Ξ΅ If the above is true, the following also must be true: 9 Γ— (1 - 0.999999...) = Ξ΅ Γ— 9 Let's calculate: 0.999... Γ— 9 = ─────────── 8.1 81 8...
What are some classic fallacies?
I keep hearing about this weird type of math called calculus. I only have experience with geometry and algebra. Can you try to explain what it is to me?
How would you describe calculus in simple terms?
Is 1 classified as a prime number? And if so, why? If not, why not?
Is 1 a prime number?
The argument against this is that 0.99999999... is "somewhat" less than 1. How much exactly? 1 - 0.999999... = Ξ΅ (0) If the above is true, the following also must be true: 9 Γ— (1 - 0.999999...) = Ξ΅ Γ— 9 Let's calculate: 0.999... Γ— 9 = ─────────── 8.1 ...
I covered hyperbolic triganomic functions in a recent maths course. However I was never presented with any reasons as to *why* (or even if) they are useful. Is there any good examples of their uses outside academia?
There came a time in mathematics when people encountered situations where they had to deal with really, really, really small things. Not just small like 0.01; but small as in *infinitesimally small*. Think of "the smallest positive number that is still greater than zero" and you'll realize what sort of problems mat...
The Weyl equidistribution theorem states that $\{n \xi\}$ is uniformly distributed for $\xi$ irrational, modulo 1; it can be proved using a bit of ergodic theory or simply playing with trigonometric polynomials and using the fact they are dense in the space of all continuous functions modulo 1. In particular, one show...
How do you prove that $p(n \xi)$ for $\xi$ irrational is uniformly distributed modulo 1?
What are some classic fallacious proofs?
Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why 0^0 should be undefined.
Why is x^0 = 1 except when x = 0?
I was playing around with the squares and saw an interesting pattern in their differences. <pre> 0^2 = 0 + 1 1^2 = 1 + 3 2^2 = 4 + 5 3^2 = 9 + 7 4^2 = 16 + 9 5^2 = 25 + 11 6^2 = 36 etc. </pre> Also, in a very related question, wh...
Why are the differences between consecutive squares equal to the sequence of odd numbers?
I have a square that's 10m x 10m. I want to cut it in half so that I have a square with half the area. But if I cut it from top to bottom or left to right, I don't get a square, I get a rectangle! I know the area of the small square is supposed to be 50, so I can use my calculator to find out how long a side should ...
how do i cut a square in half?
0^x = 0, x^0 = 1 both are true when x is not 0 what happens when x=0? undefined, because there is no way to chose one definition over the other. Some people define 0^0 = 1 in their books, like Knuth, because 0^x is less 'useful' than x^0.
This is a question of definition, the question is "why does it make sense to define x^0=1 except when x=0?" or "How is this definition better than other definitions?" The answer is that $x^a * x^b = x^{a+b}$ is an excellent formula that makes a lot of sense (multiplying a times and then multiplying b times is the sa...
I was playing around with the squares and saw an interesting pattern in their differences. <pre> 0^2 = 0 + 1 1^2 = 1 + 3 2^2 = 4 + 5 3^2 = 9 + 7 4^2 = 16 + 9 5^2 = 25 + 11 6^2 = 36 etc. </pre> (Also, in a very related question, w...
By matrix-defined, I mean <pre> < a b c > x < d e f > = | i j k | det(| a b c |) | d e f | </pre> (instead of the definition of the product of the magnitudes multiplied by the sign of their angle, in the direction orthogonal) If I try cross producting two vectors with no `k` component, I get one...
It seems like functions that are continuous always seem to be differentiable, to me. I can't imagine one that is not. Are there any examples of functions that are continuous, yet not differentiable? The other way around seems a bit simpler -- a differentiable function is obviously always going to be continuous. B...
Are there any functions that are (always) continuous yet not differentiable? Or vice-versa?
To understand the benefits of using quaternions you have to consider the alternatives. Here are few ways with a summary of the pros and cons: - Euler angles - Rotation matrices - Axis angle - Quaternions - Rotors (normalized Spinors) Euler angles are the best choice if you want a user to specify an...
To understand the benefits of using quaternions you have to consider different ways to represent rotations. Here are few ways with a summary of the pros and cons: - Euler angles - Rotation matrices - Axis angle - Quaternions - Rotors (normalized Spinors) Euler angles are the best choice if you want...
Complex numbers involve the square root of negative one, and most non-mathematicians can't accept that such a number exists. What's the best way to explain that complex numbers really do exist, and are just as "valid" as real numbers?
Do complex numbers really exist?
I've come across statements in the past along the lines of "function f has no closed form integral", which I assume means that there is no combination of the operations: * addition/subtraction * multiplication/division * raising to powers and roots * trigonometric functions * exponential functions * logarithmic...
How can you prove that a function has no closed form integral?
The argument isn't worth having, as you disagree about what it means for something to 'exist'. There are many interesting mathematical objects which don't have an obvious physical counterpart. What does it mean for the Monster group to exist?
The concept of mathematical numbers and "existing" is a tricky one. What actually "exists"? Do negative numbers exist? Of course they do not. You can't have a negative amount of apples. Yet, the beauty of negative numbers is that when we define them (rigorously), then all of a sudden we can use them to solve p...
I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is "the set of all sets" a paradox? It seems like it would be fine, to me. There is nothing paradoxical about a set con...
Why is "the set of all sets" a paradox?
I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $frac{4}{3}$ is so random! How could somebody guess something like this for the formula?
Why is the volume of a sphere $\frac{4}{3}\pi r^3$?
There are a few good answers to this question, depending on the audience. I've used all of these on occasion. **A way to solve polynomials** We came up with equations like `x - 5 = 0, what is x?`, and the naturals solved them (easily). Then we asked, "wait, what about `x + 5 = 0`?" So we invented negative numbers...
Complex numbers involve the square root of negative one, and most non-mathematicians can't accept that such a number exists. What's the best way to explain *to a non-mathematician* that complex numbers really do exist, and are just as "valid" as real numbers?
Calculus is the mathematics of change. In algebra, almost nothing ever changes. Here's a comparison of some algebra vs. calc problems: algebra: car A is driving at 50 kph. How far has it gone after 6 hours? calc: car B starts at 10 mph and begins accelerating at the rate of 10 kph^2 (kilometers per hou...
Does this give you any ideas? ![alt text][1] [1]: http://farm5.static.flickr.com/4119/4813793520_0d7fa53b99_o.png
Consider the task of generating random points uniformly distributed within a circle of a given radius $r$ that is centered at the origin. Assume that we are given a random number generator $R$ that generates a floating point number uniformly distributed in the range $[0, 1)$. Consider the following procedure: 1....
Will this Procedure Generate Random Points Uniformly Distributed within a Given Circle? Proof?
We will will first consider the most common definition of i, as the square root of -1. When you first hear this, it sounds crazy. 0 squared is 0; a positive times a positive is positive and a negative times a negative is positive too. So there doesn't actually appear to be any number that we can square to get -1. A ...
I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to *why* (or even if) they are useful. Is there any good examples of their uses outside academia?
I have a collection of 3D points in the standard x, y, z vector space. Now I pick one of the points `p` as a new origin and two other points `a` and `b` such that `a - p` and `b - p` form two vectors of a new vector space. The third vector of the space I will call `x` and calculate that as the cross product of the firs...
To understand the benefits of using quaternions you have to consider different ways to represent rotations. Here are few ways with a summary of the pros and cons: - Euler angles - Rotation matrices - Axis angle - Quaternions - Rotors (normalized Spinors) Euler angles are the best choice if you want...
Basic definitions: a tiling of d-dimensional Euclidean space is a decomposition of that space into polyhedra such that there is no overlap between their interiors, and every point in the space is contained in some one of the polyhedra. A vertex-uniform tiling is a tiling such that each vertex figure is the same: eac...
I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is "the set of all sets" a paradox? It seems like it would be fine, to me. There is nothing paradoxical about a set con...
I'm going to use $P(a,b)$ for probability of x = a, and y = b. $P(a,b) = P[x = |a|]P[y = |b|] * 1/4$ for any a, b inside the square. Therefore you are picking a coordinate uniformly inside the square. Now the problem reduced to the following: Pick a element uniformly in a set A, then discard ones that are n...